Table of integrals

Integration and finding antiderivatives is one of the two basic operations in calculus and since it, unlike differentiation, is non-trivial, tables of known integrals are often useful. Here is the beginning of such a table.

We use C for an arbitrary constant that can only be determined if something about the value of the integral at some point is known.

$\int x^{n}\,dx={\frac {x^{n+1}}{n+1}}+C\qquad {\mbox{ if }}n\neq -1$

$\int x^{-1}\,dx=\ln {\left|x\right|}+C$

---

$\int \ln {x}\,dx=x\ln {x}-x+C$

---

$\int e^{x}\,dx=e^{x}+C$

$\int a^{x}\,dx={\frac {a^{x}}{\ln {a}}}+C$

---

∫(1+x²)^-1\,dx = arctan(x) + C

∫(1-x²)^-1/2\,dx = arcsin(x) + C

∫x(x²-1)^-1/2\,dx = arcsec(x) + C

---

∫cos(x)\,dx = sin(x) + C

∫sin(x)\,dx = -cos(x) + C

∫tan(x)\,dx = -ln|cos(x)| + C

∫csc(x)\,dx = -ln|csc(x)+cot(x)| + C

∫sec(x)\,dx = ln|sec(x)+tan(x)| + C

∫cot(x)\,dx = ln|sin(x)| + C

---

∫sec²(x)\,dx = tan(x) + C

∫csc²(x)\,dx = -cot(x) + C

∫sin²(x)\,dx = x/2-(sin(2x))/4 + C

∫cos²(x)\,dx = x/2+(sin(2x))/4 + C

---

∫sinh(x)\,dx = cosh(x) + C

∫cosh(x)\,dx = sinh(x) + C

∫tanh(x)\,dx = ln(cosh(x)) + C

∫csch(x)\,dx = ln|tanh(x/2)| + C

∫sech(x)\,dx = arctan(sinh(x)) + C

∫coth(x)\,dx = ln|sinh(x)| + C

These formulas only state in another form the assertions in the Table of Derivatives.

Definite integrals

There are some functions whose antiderivatives cannot be expressed in closed form. However, the values of the definite integrals of these functions over some common intervals can be calculated. A few useful definite integrals are given below.

$\int _{0}^{\infty }{{\sqrt {x}}\,e^{-x}\,dx}={\frac {1}{2}}{\sqrt {\pi }}$

$\int _{0}^{\infty }{e^{-x^{2}}\,dx}={\frac {1}{2}}{\sqrt {\pi }}$

$\int _{0}^{\infty }{{\frac {x}{e^{x}-1}}\,dx}={\frac {\pi ^{2}}{6}}$

$\int _{0}^{\infty }{{\frac {x^{3}}{e^{x}-1}}\,dx}={\frac {\pi ^{4}}{15}}$

<math>\int_0^\infty\frac{\sin(x)}{x}\,dx=\frac{\pi}{2}